An imprcsn’on of Higbie’s suggested mechanism f m mass transfer across an interface via eda) surface renewal.
Liquid Film Controlled Mass Necessary improvement in scale-up designs for agitated reactors may come from using diffusivity as a correlating parameter and better description of the rate controlling step. D. N. MILLER rates for processes carried out in agitated Reaction vessels are frequently governed by interphase mass transfer mechanisms and the rate controlling step is often mass transport across a liquid film. I n this review we will consider the theory and application of currently available methods for the important case of liquid filmcontrolled mass transfer. Theoretical Develo~menb
The various mass transfer mechanisms which have been postulated to fit experimental data are based on Fick’s laws for diffusion processes. Through Fick‘s first law, which is applicable for steady state conditions, the following relationship is obtainable:
N
-
k (c,
- c)
(1)
The assumption implicit in the use of the difference (c, - c) as a driving force is that a saturation concentration always exists at the phase interface. Although this assumption is common in mass transfer work, it is not always valid. I t is possible, where mass transfer rates are high, for the interface solute concentration to fall short of saturation. When this occurs, the driving force is not so great as the indicated difference c. c.
-
18
INDUSTRIAL AND ENGINEERING CHEMISTRY
The Film Theory. The rate constant k in Equation 1 can be interpreted as the ratio of molecular diffusivity to a stagnant film thickness:
D k=-
F
(2)
This is the basic concept of the film theory proposed by Lewis and Whitman in 1924 (39). By this concept, mass transfer into or out of a moving fluid phase occurs by molecular diffusion through a thin film of stagnant fluid at the phase boundary. Mass transfer through the stagnant film is presumed to be slower by orders of magnitude than in the bulk phase and controls the overall single phase rate. As indicated in Equation 2, the rate of mass transfer is directly proportional to molecular diffusivity and diffusion is assumed to occur only in a direction perpendicular to the interface. With the possible exception of mass transfer from spheres of solute suspended in a stagnant solvent (38), the film theory has never been totally adequate for any transfer process. One of its most serious limitations is that the mass transfer rate is presumed to be constant with time, which follows from Fick’s first law. In many
1 hc illustration dejiicts solute ban&
between Iddies in
a
successan of steps during a siqh contact
"..... ,>*.+:',:..
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Interfacial models based on discontinuous contact have become accepted
Danckwerts’ Modification. Higbie’s assumption of a constant t, is improbable unless the physical nature of the contacting device imposes this constraint on the system. Danckwerts proposed a modification of the penetration theory which makes this assumption unnecessary (74). He pointed out that any irregular, discontinuous contacting of a phase interface by a fluid can be characterized by a distribution of contacting times + ( t ) . C#J represents the fraction of fluid elements that remains at a phase interface, for a time t. Danckwerts assumed that the displacement of fluid elements from a phase boundary was completely random, there being no dependence on time of contact. The fractional rate of displacement of fluid elements, which is, therefore, a constant over all contact times, will be calledf. For any small increment in contact time dt, the fraction of fluid elements entering the “age group” t to (t d t ) from the “age group” ( t - d t ) to t is equal to +(t) dt. This can be equated to the fraction of fluid elements entering the “age group” ( t - dt) to t , less the fraction displaced in the time interval dt:
+
C#J(t)dt= +(t - d t ) dt (1 - f dt)
The term, +(t
(6)
- d t ) , can be expanded in a Taylor series.
Making this substitution in Equation 6, and neglecting all differentials of powers greater than one, gives d+ _ -- dt
Since
l
+(t) dt =
f d o
1, by definition, the solution to Equa-
tion 8 is
d t ) =
J
exp
(-fd
(9)
Higbie’s development was shown to give the mass flux expression (Equation 5) for fluid elements having a single exposure time t,. By Danckwerts’ concept, where there is a broad continuum of exposure times, the fraction of the fluid elements having a particular esposure time t, is + ( L e ) dt,. Equation 5 multiplied by +(t,) dt, represents the contribution to the mean mass flux of mass transfer occurrine; in fluid elements of exposure time t,
Integration of Equation 10 over the full range of exposure times from zero to infinity gives Danckwerts’ expression for the mean mass flus :
m = .\/of 20
(6,
- c,)
(11)
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
As both Higbie’s Equation 5 and Danckwerts’ Equation 11 indicate, the penetration theory predicts that rate of mass transfer is proportional to the square root of molecular diffusivit y , Danckwerts originally conceived his penetration model as a mechanism for mass transfer in turbulent fluid systems. I n a turbulent field, there is a continuous circulation of eddies. These can be pictured as attaching and displacing at a phase interface periodically, mass transfer occurring during the intervals of contact. Through the dependency of rate on D1/2,the penetration theory has been shown to apply in certain types of contactors in which the bulk fluid streams are in turbulent flow. Examples are both gas and liquid phase mass transfer in bubble-cap columns (7): and liquid phase mass transfer in packed columns (34). Successful applications of the penetration theory have also been made in certain types of contactors where laminar liquid films are exposed to mass transfer for brief, measurable intervals. This has been done, for example, with short, wetted-wall columns (62) and liquid jets (47). The same principle has been advanced to explain the applicability of penetration model to liquid phase mass transfer in packed columns (75). I t has been suggested that the liquid is distributed in laminar films over the wetted surfaces of the packing, that mass transfer occurs along these exposed surfaces, and that complete liquid phase mixing occurs in the spaces between packing. Boundary Layer Theory. A further refinement to the penetration model can be made through the application of boundary layer theory. One of Higbie’s assumptions was that, when mass transfer occurs in a fluid element in contact with an interface, the fluid element remains static. In many types of contactors, it seems more probable, however, that a fluid velocity component exists along the interface in the direction transverse to mass transfer. For this situation, the time variable in Equation 3 can be expressed in terms of distance y and velocity u in the direction parallel to the interface :
For the case of flow along a flat surface, Blasius has shown that the v-elocity parallel to the surface in a laminar boundary layer is a function of xdu,//v3’, where u is the bulk fluid LTelocity, and v is the kinematic viscosity (56). From the surface out to a point in the boundary layer where velocity u is about half u,, this relationship is approximately linear. AUTHOR
D.
zng Department
of
Miller is a chemical engzneer in the EngineerE. I . du Pont de .lTemours 3 Co.
_U -- 0.33 uw
xi;
If it can be assumed that, during the period of mass transfer, solute does not penetrate into the boundary layer beyond the point where u/u, = 0.5, the ratio u / u , can be substituted for u in Equation 12. This results in a partial differential equation relating the variables c, x , and y, rather than c, x , and t as was the case in Higbie’s development. This partial differential equation can be solved for the gradient & / a x (46). When this expression is evaluated at x = o, multiplied by molecular diffusivity, and averaged over the distance of contact y = 1, the result is
fl =
0.852
the sphere. The outer portions of the loops follow, in the reverse direction, the contour of the surface of tlie sphere. Rotated around the polar axis, the closed streamlines form concentric tori. Conkie and Savic (72) have made a theoretical analysis of the effects of circulation in bubbles and drops on external mass transfer. They show that the thickness of the external boundary layer is dependent on the circumferential velocity of the fluid in a bubble or drop. Circumferential velocity in turn is a function of the Reynolds number N,,, and the viscosities p and pi of the external and internal fluids, respectively. The expression relating these variables is :
$ (?)>”’ - c,) (c,
Rate of mass transfer, as predicted by boundary layer theory, is proportional to the two-thirds power of molecular diffusivity. This dependency actually follows from the assumed linear relationship between u and x only. I t is not dependent on any relationship that may exist between u and y. Although the flat plate solution has been used to illustrate the application of boundary layer theory, the two-thirds exponent has also been derived for laminar boundary layers along spherical and cylindrical surfaces. Boundary layer theory has been shown to apply, through the dependency of the rate on D2I3,to gas phase mass transfer in packed columns (58). A rate correlation based on boundary layer theory for both gas and liquid phase mass transfer in packed columns has also been reported (9). Mass transfer from single, fixed spheres of solute immersed in moving fluid media follows the predictions of boundary layer theory (22, 50, 57). The same is true of evaporation from pure liquid drops falling in air (27, 57). The general case of mass transfer from drops or bubbles moving through fluids is much more complex. Bubbles and Drops. Small bubbles and drops in dense clusters behave as rigid spheres (8). Larger, widely dispersed bubbles and drops, however, tend to deform and oscillate in moving through fluids. Also, viscous forces between the fluid phases create iniernal circulation currents. Very little is known about deformation and oscillation. Theoretical work on large bubbles and drops has been focused, for the most part, on the effects of internal circulation. I n order to make the problem mathematically tractable, certain assumptions have customarily been made. The bubbles or drops have usually been assumed to be perfect spheres; internal circulation has usually been assumed to follow Hadamard’s idealized viscous flow pattern (37). Hadamard’s streamlines are closed, concentric loops bounded by the surface of the sphere and the polar axis, this being the axis oriented in the direction of translation of the sphere. Inside the sphere, the streamlines run parallel to the polar axis in the direction of gross movement of
{I - [I
(~)>’]’’>>”’ (15)
+
where 6” = 6/6,, the true boundary layer thickness 6 referred to a hypothetical thickness 6, for zero circumferential velocity hiRes= duw/v, the Reynolds number for a sphere of diameter d When the term N,,: ( p / p J 2 is small, the circumferential velocity is small and 6” is approximately one. This is the case for liquid drops falling through a gas. A substantial boundary layer then develops and mass transfer rate is proportional to (27, 50). When NRea ( p / p J 2 is large, the circumferential velocity approaches the main stream velocity and 6* is approximately zero. A bubble rising through a liquid exhibits this characteristic. At the extreme where the boundary layer is negligibly small, the velocity gradients established by potential flow extend to the interface. The potential flow gradients can be used in calculating the external mean mass flux (53). Rate of external mass transfer predicted from potential flow theory is proportional to D112. Experimental evidence of this behavior for single bubbles rising through liquids is inconclusive. External mass transfer rates for single liquid drops in moving liquid media, however, have been found to vary with D1I2where there is rapid movement of the drop surface (25). Where the drop surface is slowly moving or immobile, the D213dependency applies. I n characterizing mass transfer inside drops, steady state diffusion in a stagnant fluid sphere is usually used as a reference model. For the extraction of a solute from a stationary drop into a dilute surrounding fluid where external resistance to mass transfer is negligible : _ M_ ( t_ ) - - x6 ; e x 1 p-
M(o)
7rz n = l
n
(
4n2,Dt) ~
(16 )
where
M ( t ) = mass of solute remaining in the drop at time t M(o) = initial mass of solute in the drop Kronig and Brink (36) have developed the mathematics for extraction of solute from a drop with internal VOL. 5 6
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circulation, where the drop is moving under the influence of gravity through another fluid. Their model assumes that molecular diffusion occurs normal to Hadamard’s streamlines. Circulation of the streamlines relative to diffusion is assumed to be rapid, so that concentration along each streamline is essentially constant. Their equation, for a dilute surrounding fluid and negligible external mass transfer resistance is :
M ---=-E (t) 3 A2expM(o) 8n-1 where A1 = 1.32, A
2 =
> ”)
(64 ___
A, = 1.678, Xz = 9.83
.4rough idea of the relative rates of mass transfer in these two models can be obtained by an application of the film theory (26). It is possible to rewrite Equation 1, for internal transfer in the spherical configuration, assuming c, = 0, as: 6k - dt (E
U,
(17)
0.73
dc -
Making this substitution along with ATsc = u/‘D in Equation 14, the boundary layer theory mass flux dependency can be expressed in terms of the dimensionless Reynolds and Schmidt numbers. Extracting the terms which collectively represent the mass transfer rate constant in this expression, the following equation can be derived : k N-,, = - = 0,852 i\’Re-l/* ly,qc-2iY (23)
C
From Equation 18 :
The dimensionless ratio k / u called the Margoulis number, is analogous to the Stanton number in heat transfer. If both sides of Equation 23 are multiplied by the product of the Reynolds number and the Schmidt number, the more familiar ‘\TSh =
kl
-
D
is obtained. The dimensionless ratio kl,;D is the Sherwood number. The form of Equation 24 can be derived independently by dimensional analysis :
ATSl, The first terms in Equations 16 and 17 can be used to approximate the series solutions. If the exponents of these terms are compared with the exponent of Equation 19, the following rate constant relations analogous to Equation 2 are apparent: for the rigid sphere
and for the Kronig and Brink model
Experimental internal drop transfer rates have been found that vary from the rigid sphere characteristic for highly viscous liquids up to three or four times the Kronig and Brink rate for fluids of low viscosities (7, 45). Other approaches have also been attempted in characterizing internal drop transfer. Equation 1 6 has been used with an “effective” rather than a molecular diffusivity (26). The penetration theory has been applied to both internal and external drop mass transfer, assuming the time of exposure t, to be the time for a drop to displace one diameter (63). Both the theoretical basis and the experimental support for this approach are unsatisfactory, however. The Gilliland-Sherwood Correlation. The mean mass flux relationship developed from boundary layer theory involves a characteristic distance of contact, I , over which mass transfer occurs. This characteristic length can be expressed as a function of the Reynolds number :
22
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
- 0.852 I V ~ ~ ! ~ ~ \ ~ (24) , ~ ~ ~ ~
=
(25)
C.YR;:lTS:
Known as the Gilliland-Sherwood correlation: this relationship was first applied to the evaporation of water and various organic liquids into a turbulent air stream in a wetted-wall c.olurnn (23). It has since been used in numerous mass transfer correlations. The three constants are typically evaluated by fitting Equation 25 to experimental data. The exponent b is often cited in support of one of the three mass transfer mechanisms. Values of zero, +1/2, and +1/3 correspond to the film, penetration, and boundary layer theories, respectively. Applications of Theory in the Correlation of Data
Several correlations of data on liquid film-controlled mass transfer in agitated vessels have recently been reported. Each employs a different approach, but shows conformity with the boundary layer theory through the characteristic dependence of rate on D2j8. Marangozis and Johnson (42) have made use of the Gilliland-Sherwood equation to correlate both their own data and data extracted from the literature. Their versus is reproduced in plot of the ratio ATSh/L\TS~‘B Figure 1. The points on their plot represent data taken in several types of unbaffled, agitated contacting systems. These include cylinders of solute suspended in solvent, either the containing vessels (5) or the cylinders being rotated (35, 57), rotating cylindrical electrodes suspended in solutions of ions undergoing either oxidation and agitated vessels in which or reduction reactions (17), pellets of solute were freely suspended (27, 29). The mean deviation of the experimental points from 16.7%. the least-mean-squares straight line fit is The sizes of the agitated systems employed were about 0.01 gal. in the case of the rotating vessels, 0.2 to 0.8 gal. for the rotating cylinders and electrodes, and 0.7 to 350 gal. for the suspended solids. A similar correlation was
*
developed for baffled systems. The Nsh/Ns,liaordinates, compared at the same NRe values, were about 3770 lower in the baffled, compared with the unbaffled, systems. These are semiempirical applications of the GillilandSherwood correlation. Certain of the correlating parameters were judiciously selected to get the best possible fit. The length parameter in Nsh and N,, was taken as the hydraulic diameter of the annulus in the case of the rotating vessels; the gap width in the case of the rotating cylinders and electrodes; and the impeller diameter in the case of the suspended pellets. T h e velocity term in N,, was taken as the peripheral speed of the rotating elements in the systems-the rotating vessels, cylinders, electrodes, or impellers. T o be theoretically correct, the length parameter in NShand N R a should be distance along the phase interface where boundary layer development and mass transfer occur without interruption. The bulk fluid velocity past the phase interface should be the velocity term used in NRe. A variation on the semiempirical use of the GillilandSherwood relationship is reported by Calderbank and Moo-Young (8). In this development, the idealized concept of a homogeneous isotropic field is used to characterize mass transfer in turbulent systems. A Reynolds number is derived (3, 24) by taking the ratio of dynamic pressure due to turbulence, [d2/3(6.75X g P / V ) 2 / ap113] to viscous shear stress [$I2 (6.75 X g P/ V)' 12 1. 0.202 g'iR d213 (P/V)'/6 T\:Re = -(26) ,I/% $/e where
g = gravitational acceleration, ft./hr.2 P = power dissipated, hp. V = volume,gal. X lo3 This Reynolds number is then used in the GillilandSherwood correlation. I n work with sieve plate columns, these authors found no evidence of a n effect of bubble size on the mass transfer rate constant. There have been several semi-
,041
...
1
~
,......
"_
1
.
"
I
.,.. .... .......
I,
.
..
empirical correlations for agitated vessels, including the one previously discussed, which have been fairly successful without the inclusion of a size parameter for the discontinuous phase. On the strength of these observations, Calderbank and Moo-Young have chosen to eliminate this size parameter in their correlation. The Schmidt number exponent b, however, is taken to be 1/3, as would be predicted by boundary layer theory. With the elimination of particle, drop, or bubble size, along with the assumption of b = 1/3, Calderbank and Moo-Young's adaptation of the Gilliland-Sherwood correlation reduces to (P/ V )v '14 k = C' 2vsc -2P (27)
[y]
The analogous relationship for heat transfer is
L
P
J
where h = heat transfer coefficient, p.c.u./(lb.) ("C.) = liquid density, lb./cu. ft. p N p , = c p p / k H = Prandtlnumber k H = liquid thermal conductivity, p.c.u./ (hr.) (ft.) ("C.) These equations have been used by Calderbank and Moo-Young to correlate data taken from the literature on mass and heat transfer in baffled, agitated vessels. The power dissipation term P was obtained by measuring agitator power input in the reconstructed equipment, where this information was not provided in the original reference. The Calderbank and Moo-Young plot of k or h / c p p versus [ ( P / V ) V / P ' N2issc / ~ ] OT P r is reproduced in Figure 2. C.G.S. units are employed. The mean deviation of the experimental points from the straight line fit is not given. By inspection, however, the fit looks almost as good as that of the Marangozis-Johnson correlation, this being achieved over a 107-foldrange of mass and heat transfer coefficients. The mass transfer data which are plotted were taken in systems ranging in size from 0.3 to 5.8 gal.; the heat transfer data were obtained in systems ranging from 5.8 to 750 gal. in size. T h e mass transfer measurements were made on processes involving the dissolution of a solute which was either in the form of freely suspended particles (30, 32, 33, 43, 49) or cast in a ring depression in the bottom of the retaining vessel (67). The heat transfer measurements were made by either heating or cooling the contents of the vessels with coils (55) and jackets (6, 77). In their sieve plate studies, Calderbank and MooYoung evaluated the power dissipated in passing a gas through a liquid from the relationship
P/V
=
6.75 X
u, p
(29)
where u, = superficial velocity, ft./hr.
Re
Figure 7 . Marangozis and Johnson correlation for mass transfer in unbaJi'ed, agitated vessels, iepoduced with permission
Their sieve plate mass transfer data taken alone show no evidence of a dependence on power input per unit volume. They do fall within the scatter of experimental points on Figure 2, however. VOL. 5 6
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1 o2
D, (microns)
+
